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Indefinite integralsCalculusCollegeabout 0:56

Integrate 1/(2x + 1) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: ln(abs(2 x + 1)) / 2 + C.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int \frac{1}{2x+1}\,dx$
2. Step by step
1. $\int \frac{1}{2x+1}\,dx$
  Start with the integral
2. $\int {\left(2x+1\right)}^{-1}\,dx = \frac{\ln\left(\left|2x+1\right|\right)}{2}$
  Linear substitution u = ax + b: ∫ 1/u du = ln|u|, divide by a
3. $\int 1 {\left(2x+1\right)}^{-1}\,dx = \frac{\ln\left(\left|2x+1\right|\right)}{2}$
  Constant multiple rule: ∫ k·f dx = k∫ f dx
4. $\int \frac{1}{2x+1}\,dx = \frac{\ln\left(\left|2x+1\right|\right)}{2} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{\ln\left(\left|2x+1\right|\right)}{2} + C$
∫ 1 / (2 x + 1) dx = ln(abs(2 x + 1)) / 2 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate 1/(2x plus 1) dx. Find the indefinite integral (antiderivative) of 1 / (2 x plus 1) with respect to x.

Step by step. Start with the integral. Linear substitution u equals ax plus b: the integral of 1/u du equals ln|u|, divide by a. Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Simplify and add the constant of integration.

The answer. So the answer is ln(abs(2 x plus 1)) / 2 plus C. An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

Check. Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.

About indefinite integrals

Find antiderivatives with standard rules, substitution and integration by parts.

Integrate 1/(2x + 1) dx: Animated Step-by-Step Lesson